MANY-BODY EFFECTS IN BOSE-EINSTEIN CONDENSATES OF DILUTE ATOMIC GASES BRETT DANIEL ESRY. B.S., Kansas State University, 1993

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1 MANY-BODY EFFECTS IN BOSE-EINSTEIN CONDENSATES OF DILUTE ATOMIC GASES by BRETT DANIEL ESRY B.S., Kansas State University, 1993 M.S., University of Colorado, 1996 A thesis submitted to the Faculty of the Graduate School of the University of Colorado in partial fulfillment of the requirements for the degree of Doctor of Philosophy Department of Physics 1997

2 This thesis for the Doctor of Philosophy degree by Brett Daniel Esry has been approved for the Department of Physics by Chris H. Greene Eric Cornell Date The final copy of this thesis has been examined by the signators, and we find that both the content and the form meet acceptable presentation standards of scholarly work in the above mentioned discipline.

3 Esry, Brett Daniel (Ph. D., Physics) Many-body Effects in Bose-Einstein Condensates of Dilute Atomic Gases Thesis directed by Professor Chris H. Greene The recent experimental achievement of Bose-Einstein condensation in a dilute alkali gas has spurred a great deal of interest among physicists from many fields. Dilute atomic gas experiments are particularly attractive, compared to experiments on the closely related phenomena of superfluidity and superconductivity, because a dilute gas is a weakly interacting system which is far more amenable to theoretical description. Experimentally, dilute gas experiments are advantageous because relatively straightforward and convenient diagnostics exist, using laser excitation of atomic transitions. As a result, dilute atomic gas experiments can be more completely understood using first principles theoretical treatments. I have adapted the Hartree-Fock, random phase, and configuration interaction approximations to describe systems of interacting bosons, and have shown that such systems can be treated accurately and efficiently within a particle number conserving approximation. In fact, the resulting approximations are remarkably similar to those made in the standard Bogoliubov approach and lead to largely the same equations. A key conclusion is that a system of interacting bosons can be treated in a manner analogous to that used to describe the electronic states of atoms. The hope is that the knowledge and intuition that have been gained from the extensive study of the atomic structure problem will ultimately lead to a deeper understanding of the quantum mechanical states of interacting, trapped atoms. In the course of this work, several phenomena are studied using both the Hartree-Fock approximation and the random phase approximation. The resulting analysis of the stability criteria for single and double condensates improves on results available in the literature in both cases. The double condensate ground state is explored for various hyperfine and isotopic combinations of rubidium in fully three-dimensional configurations for realistic numbers of atoms. Random phase approximation excitation spectra are also calculated for both single and double

4 iv condensates. Many of these predictions have not yet been tested experimentally, nor is there any other theoretical treatment with which comparisons can be made. A systematic study of spatial symmetry breaking at the Hartree-Fock level of approximation for the ground state of double condensates is also presented.

5 ACKNOWLEDGEMENTS I would first like to thank my advisor Chris Greene for his guidance and support during my years at JILA. I have gained both greater physics knowledge and a broader perspective on the field under his supervision. This bulk of this work, in fact, grew from his suggestion a little less than two years ago that the atom trapping experiments underway at JILA could be modeled using techniques from atomic structure. Since then, we have come a long way not only in our understanding of systems of interacting bosons but also in our understanding of the subtle differences between our approach and conventional approaches. Many outstanding issues remain on both fronts, however, that I hope can be resolved through our continued collaboration. Second, I would like to acknowledge the benefits of working in JILA where active atomic physics programs especially Bose-Einstein condensation are being pursued. Within Chris group, I have had many useful conversations with Jim Burke and John Bohn both on two-body scattering and the many-boson problem. From the greater Bose-Einstein condensation effort here at JILA, many people have posed questions or offered suggestions that have deepened my understanding of the field. I would like to mention a few here with whom I have had the most interaction: Jinx Cooper, Eric Cornell, Carl Wieman, Murray Holland, Jason Ensher, Mike Matthews, and Debbie Jin. In particular, I would like to recognize the many constructive criticisms of the manuscript of this thesis by Eric Cornell and Carl Wieman. Third, I would like to acknowledge initial discussions on this project with Mike Cavagnero and discussions on the basics of some of the methods presented here with Jim Shepard. I want to acknowledge the constructive comments of Peter Zoller on a portion of this manuscript as well as the many helpful suggestions of Leo Radzihovsky on the entire manuscript. Finally, I would like to acknowledge the financial support of this research by the Department of Energy, Office of Basic Energy Sciences.

6 CONTENTS CHAPTER 1 INTRODUCTION STANDARD APPROACHES FOR BOSE-EINSTEIN CONDENSATES Review of second quantization Bogoliubov approximation Thomas-Fermi approximation ZERO TEMPERATURE THEORY FOR SINGLE COMPONENT CONDEN- SATES Hartree-Fock approximation First-quantized approach Second-quantized approach Time-dependent Hartree-Fock equation Stability of the Hartree-Fock solution Random phase approximation Time-independent derivation Time-dependent derivation Configuration Interaction Pseudopotential approximation Adiabatic hyperspherical approximation Hartree-Fock with Morse interactions Hartree-Fock in the shape independent approximation Isotropic trap results Anisotropic trap results

7 vii 4 ZERO TEMPERATURE THEORY FOR MULTIPLE COMPONENT CON- DENSATES Hartree-Fock approximation First-quantized approach Second-quantized approach Time-dependent Hartree-Fock equations Stability of the Hartree-Fock solution Random phase approximation JILA baseball trap results Rb+ 87 Rb Rb Rb TOP trap results , 2 + 1, , 1 + 1, Spatial symmetry breaking SUMMARY BIBLIOGRAPHY APPENDIX A ADIABATIC HYPERSPHERICAL APPROACH B GENERAL MANY-BODY HAMILTONIAN MATRIX ELEMENT C BASIS SPLINES D SOLVING THE HARTREE-FOCK EQUATION D.1 Iterative diagonalization D.2 Imaginary time propagation D.2.1 Finite differences D.2.2 Basis splines E RELATED PUBLICATIONS

8 FIGURES FIGURE 1 Variational total energy per particle for an isotropic trap Critical value of α as a function of trap anisotropy Rb+ 87 Rb scattering length Ground state orbital energy for an isotropic trap Total ground state energy for 87 Rb in an isotropic trap Ground state orbitals for an isotropic trap Effective one-body potentials for an isotropic trap Comparison of quasi-hartree-fock and Hartree-Fock for an isotropic trap L=0 excitation spectrum for an isotropic trap Ground Hartree-Fock energies for 87 Rb in a TOP trap Ground state energy shift for 87 Rb in a TOP trap Even z-parity spectrum for 87 Rb in a TOP trap Odd z-parity spectrum for 87 Rb in a TOP trap Comparison of theoretical and experimental excitation frequencies for 87 Rb in a TOP trap Critical interspecies scattering length for concentric isotropic traps Orbital energies for 85 Rb+ 87 Rb in the JILA baseball trap Ground state densities for 85 Rb+ 87 Rb in the JILA baseball trap Ground state densities for 87 Rb 2, 2 + 1, 1 in the JILA baseball trap Orbital energies and average positions for 87 Rb 2, 2 + 1, 1 in the JILA baseball trap Average positions for 87 Rb 2, 2 + 1, 1 in the JILA baseball trap

9 ix 21 Lifetimes of 87 Rb 2, 2 + 1, 1 in the JILA baseball trap Excitation spectra for 87 Rb 2, 2 + 1, 1 in a TOP trap Density oscillations for the lowest two modes of 87 Rb 2, 2 + 1, 1 in a TOP trap Ground state densities for 87 Rb 2, 2 + 1, 1 in a TOP trap Density oscillations for the ν=6 and ν=7 modes of 87 Rb 2, 2 + 1, 1 in a TOP trap Ground orbital energies for 87 Rb 2, 2 + 1, 1 in a TOP trap Excitation spectra for 87 Rb 2, 2 + 1, 1 in a TOP trap Ground state densities for 87 Rb 2, 2 + 1, 1 with an attractive interspecies interaction Density oscillations for 87 Rb 2, 2 + 1, 1 with an attractive interspecies interaction Ground state densities for 87 Rb 2, 1 + 1, 1 in a TOP trap Excitation spectra for 87 Rb 1, 1 + 2, 1 in a TOP trap Ground state densities for 87 Rb 1, 1 + 2, 1 with a 12 =107 a.u. in a TOP trap Ground state densities for 87 Rb 1, 1 + 2, 1 with a 12 =110 a.u. in a TOP trap Ground state densities for 87 Rb 1, 1 + 2, 1 with a 12 =130 a.u. in a TOP trap Excitation spectra for 87 Rb 1, 1 + 2, 1 in a TOP trap Ground state energies for 87 Rb 1, 1 + 2, 1 in an inverted TOP trap Ground state densities for 87 Rb 1, 1 + 2, 1 in an inverted TOP trap for a 12 =107 a.u Ground state densities for 87 Rb 1, 1 + 2, 1 in an inverted TOP trap for a 12 =107.1 a.u Ground state densities for 87 Rb 1, 1 + 2, 1 in an inverted TOP trap for a 12 =107.5 a.u

10 x 40 Ground state densities for 87 Rb 1, 1 + 2, 1 in an inverted TOP trap for a 12 =115 a.u Average positions for 87 Rb 1, 1 + 2, 1 in an inverted TOP trap Excitation spectra for 87 Rb 1, 1 + 2, 1 in an inverted TOP trap Coordinate definitions for the three-body problem Basis splines generated from a linear mesh Structure of basis splines matrix for two-dimensional problem Example of iterative diagonalization convergence Ground state orbital evolution in imaginary time propagation

11 xi TABLES TABLE 1 Total ground state energies for three bosons in a harmonic trap Convergence of the random phase approximation Convergence of the RPA spectrum for 87 Rb 1, 1 + 2, 1 in an inverted TOP trap Convergence of anisotropic oscillator energies using basis splines Imaginary time propagation solution of the Hartree-Fock equation

12 CHAPTER 1 INTRODUCTION The field of Bose-Einstein condensation (BEC) in a dilute alkali atom gas has emerged in the last few years as a rich and exciting commingling of several disciplines within physics. While this can often lead to conflict and confusion stemming from disparate viewpoints, it also adds to the level of excitement in the field and the variety of physics discussed. The rise of BEC has been fueled largely by the experimental observation of the phenomena first at JILA by Anderson et al. [1], then by Davis et al. at MIT [2] and Bradley et al. at Rice [3]. Since the flurry of initial observations in the summer of 1995, four more groups have reported observing BEC in an alkali gas [4, 5, 6, 7]. With several concrete experimental observations already published and the prospect of a continuous and growing stream of results emerging for the foreseeable future, theorists with backgrounds in quantum optics, atomic physics, nuclear physics, and condensed matter physics are tackling the problem with a wide variety of theoretical tools. This dissertation develops a theoretical description of a zero temperature system of interacting bosons, combining some of the viewpoints and techniques of atomic and nuclear physics. Bose-Einstein condensation, a phenomenon in which a substantial fraction of the particles in a system of bosons suddenly and spontaneously occupies the lowest quantum state as the temperature approaches absolute zero, was theoretically predicted 73 years ago [8]. Dilute atomic gas experiments have not only achieved BEC, but have also performed detailed and revealing experiments [9, 10] which have motivated several groups to develop BEC experiments and have guided other groups already actively seeking BEC. Experiments using a dilute atomic gas to study Bose-Einstein condensation are particularly attractive, compared to experiments on the closely related phenomena of superfluidity

13 2 and superconductivity, because a dilute gas is a weakly interacting system which is far more amenable to theoretical description. A dilute gas experiment is also advantageous because relatively straightforward and convenient diagnostics exist, using laser excitation of atomic transitions. As a result, dilute atomic gas experiments can be more completely understood using first principles theoretical treatments. Comparisons between theoretical and experimental results for single species condensates have shown quite good agreement for both ground and excited state properties [11, 12]. The first principles treatment of multiple component condensates can also be expected to provide good agreement with experiment. Multiple component condensates are intrinsically interesting as macroscopically occupied interacting and interpenetrating quantum states and have long been sought in low temperature physics, primarily in the 3 He- 4 He system [13]. Dilute atomic gas experiments have the advantage in addition to those already mentioned that nearly all aspects can be independently controlled to some extent. The numbers of particles present of each species, for instance, can be varied as can the amount of overlap between the condensates. The interatomic interactions can also, in principle, be varied with additional external light, magnetic, or RF fields. Theoretically, the problem of computing the zero temperature ground and excited state properties of an interacting boson system is conventionally carried out within the Bogoliubov approximation [14]. The Bogoliubov approximation abandons the strict number conservation inherent in a many-body Hamiltonian from the outset and introduces both a chemical potential and the phase of the condensate. Such analyses proceed via field theoretic methods and lead to a definition of the condensate which requires the condensate wave function to be a coherent state in the number of particles. That is, the condensate wave function is a linear combination of many ordinary Schrödinger states, each corresponding to a different number of particles. The chemical potential enters the treatment to fix the average number of particles in this mixed state. The condensate phase enters as the variable canonically conjugate to the particle number [15]. When applied to the strongly interacting system of liquid 4 He, the standard example of Bose-Einstein condensation prior to the success of dilute atomic gas experiments, this approach

14 3 leads to the concepts of phonons, maxons, and rotons [16]. In this work, I take the very different point of view that the system of bosons can be described by a strictly number conserving Schrödinger equation. A number conserving approach has been developed also by Gardiner [17], although from a very different starting point. Mann [18] also briefly investigated a number conserving approach to superfluidity using methods developed for nuclear structure. I take my cue from atomic structure methods such as Hartree-Fock and configuration interaction, and borrow techniques such as the random phase approximation from nuclear structure. Of course, both atomic and nuclear structure are concerned with systems of interacting fermions. My task, then, has been to generalize these methods to bosonic systems. A system of interacting bosons differs from a system of interacting fermions such as an atom or molecule by more than the permutation symmetry requirement. In typical experiments, an atom is essentially always in or near its ground state electronic configuration since the first excitation energy is large compared to background thermal energies. A system of bosons, on the other hand, is almost never near its true ground state configuration since background thermal energies are many orders of magnitude larger than the critical temperature of condensation. Nevertheless, experiments can now be performed in which the fractional occupation of the ground state is near unity, making the theoretical description of their zero temperature behavior worthwhile. In a strictly number conserving theory, there is a fundamental difficulty in defining the condensate. The calculated ground state is the lowest energy eigenstate of a system with exactly N bodies. As such, it is also an eigenstate of the number operator; in other words, it is a number state. As stated above, however, the standard definition of a condensate implies that the condensate is a superposition of number states for different numbers of particles. To the extent that the condensate is ordinarily defined as a coherent state, one cannot claim to calculate a condensate wave function in the number conserving scheme adopted for this dissertation. The methods developed here will, however, provide a means to calculate approximate energy eigenstates that are equally applicable or even more applicable to describe current dilute atomic

15 4 gas experiments. Of course, the thermodynamics of the system can, in principle, be wholly described given a complete knowledge of the energy spectrum. In this picture, however, at T =0 there is no depletion of the ground state. One can define quantities analogous to the ground state depletion introduced in other theories which roughly measures the strength of interparticle interactions. The difference between unity and the absolute square of the projection of the ground energy eigenstate onto the Hartree-Fock ground state, for instance, is negligible in the weakly interacting limit and large in the strongly interacting limit. For liquid 4 He, a strongly interacting system, the ground state depletion is typically reported as being on the order of 90%, meaning that 10% of the atoms are in the zero momentum state (which is not the ground energy eigenstate) at T =0 [16]. A complete construction of the energy spectrum is not a feasible task, given that condensates in the present incarnations of atomic BEC experiments contain anywhere from thousands to millions of atoms. Yet the lower lying excitations, for which the energy eigenfunctions differ little from the Hartree-Fock solution, can be calculated. Approaches such as the random phase approximation (which I develop here) and the conventional Bogoliubov transformation describe precisely these types of excitations. Both employ the familiar physical notion that approximately reduces a complex multidimensional problem to a set of uncoupled harmonic oscillators. The approximation of hopelessly complex systems by uncoupled oscillators has a long and distinguished history. It is used routinely to describe small oscillations in classical systems. The classical normal modes form the basis for classifying and analyzing nuclear vibrations in polyatomic molecules. A phonon or a photon can be regarded as an excitation quantum of a single normal mode in a system of noninteracting harmonic oscillators. The major reason for the proliferation of this approach is that it is usually the most sophisticated approximation that can be solved exactly. Despite the relative simplicity of the decoupled oscillator approach, the resulting equations are not trivial to solve for atoms in a trapping potential. Nonetheless, several groups have developed techniques for solving them. The zero temperature Thomas-Fermi or hydrodynamic

16 5 model has been developed by Stringari and coworkers [19, 20] and by Griffin and coworkers [21]. Both groups have applied this model to study the ground and excited state properties of trapped atoms. Fliesser et al. [22] and Öhberg et al. [23] were even able to completely determine the entire excited state spectrum for an anisotropic trap in the hydrodynamic limit. The Thomas-Fermi model has also been applied recently to two component condensates by Ho and Shenoy [13] to determine the ground state density and by Graham and Walls [24] to study the excitations. Variational treatments of the Gross-Pitaevskii equation have also proven informative in studies of the stability of both one and two component condensates. In particular, one component condensates have been examined by Stoof and coworkers [25, 26], by Zoller and coworkers [27], and by Fetter [28]. Zoller et al. also used their variational wave function to study the lowest few excitations [27]. More recently, they have extended their analysis to the ground and low-lying excited states of two component condensates [29]. The Gross-Pitaevskii equation has also been solved numerically by several groups for both isotropic and anisotropic traps. The first group to do so Edwards, Burnett, Clark, and coworkers [11] found that the approximate Bogoliubov transformation to a set of uncoupled harmonic oscillators yields excitation frequencies in good agreement with experiment [9]. This same group has also solved the Gross-Pitaevskii equation numerically to examine the stability of the condensate for negative scattering lengths [30] and to examine the spectroscopy of vortices [31]. You et al. [32] and Dalfovo et al. [20] have also solved the Gross-Pitaevskii equation for the low-lying excitation spectrum. The mean field approach has also been generalized to finite temperatures and the resulting equations has been solved by several groups. Öhberg et al. [33] calculated the ground state properties for temperatures below the critical temperature. Griffin et al. [34] and Dodd et al. [35] have calculated the excitation spectrum as well as the ground state properties for temperatures below the critical temperature, while Stoof et al. [36] and Stringari et al. [37] have studied how the mean field affects the critical temperature. A finite temperature mean field

17 6 study of the ground state of the two component system has even been conducted by Öhberg and Stenholm [38]. They found evidence for the interesting case of spatial symmetry breaking for a configuration of concentric trapping potentials. Of course, many aspects of BEC experiments are not described by the Gross-Pitaevskii equation. Holland, Cooper, and coworkers have studied the kinetic evolution of a dilute cloud of alkali atoms from the evaporative cooling stage through the formation of the condensate [39]. Zoller, Gardiner, and coworkers have also actively investigated the kinetic aspects of condensate formation [40]. Other groups notably Castin et al. [41], Javanainen et al. [42], and Walls et al. [43] have applied techniques borrowed from quantum optics to describe both the phase of a condensate and the interference of two condensates. In Chap. 2, I review the conventional procedure for obtaining both the equilibrium solution and the normal modes of a system of interacting bosons. The former is found by solving the Gross-Pitaevskii equation; and the latter, by solving the Bogoliubov normal mode equations. Also in Chap. 2 is a short section dealing with the notation of second quantization needed in later chapters and a short section describing the basic results of the Thomas-Fermi approximation for the ground state of identical, trapped, bosonic atoms. The next chapter, Chap. 3, develops the number conserving Hartree-Fock and random phase approximations employed throughout this work. Different derivations of each are included to make more clear the differences between and similarities with the more conventional approach in Sec Also included are first quantized derivations of the equations for these approximations that highlight the simplicity of the approximations. In Sec. 3.4, the configuration interaction method is generalized to bosons, and its connection to the random phase approximation is sketched and discussed. The remainder of the chapter is devoted to selected results for isotropic and anisotropic harmonically trapped atoms. Among the results presented is a discussion of the stability of the Hartree-Fock solution and the excitation spectrum in several regimes. Chap. 4 generalizes the treatment of Chap. 3 to the case in which more than one

18 7 species is present in the trap, and obtains Hartree-Fock and random phase approximation equations. One point that may be of particular interest is the resolution of a fundamental question about a factor of two appearing in the interspecies interaction term of the Hartree-Fock equations. Stability issues relevant to the two component system are also addressed. In the second half of the chapter, I show some of the rich varieties of phenomena that can occur in two component systems. A number of these systems are experimentally relevant. In particular, calculations are presented for the Myatt et al. [87] experimental configuration, and for a proposed experiment on mixed isotopes of rubidium (i.e. with 85 Rb and 87 Rb) that could be performed with the same apparatus. Other aspects addressed include the ground and excited state properties of a rubidium double condensate in a TOP trap. Finally, Chap. 4 concludes with a discussion of the possibility of spatial symmetry breaking in a two component system. Chap. 5 provides a brief summary and discusses possible studies for the future.

19 CHAPTER 2 STANDARD APPROACHES FOR BOSE-EINSTEIN CONDENSATES This chapter introduces the notation needed in following chapters and explains the conventional method for treating Bose-Einstein condensates. The second-quantized notation presented in Sec. 2.1 can be found in many sources, some of which are Refs. [44, 45, 46, 47, 48]. In addition, the Thomas-Fermi approximation presented in Sec. 2.3 is a simple approximation that has been used quite successfully [19, 22, 49]. The method commonly applied to zero temperature systems of bosons is developed in Sec. 2.2 for arbitrary two-body interactions. It begins with the grand canonical Hamiltonian and makes the Bogoliubov approximation for the boson field operator. This approximation breaks the particle number conserving symmetry. In fact, the solution of the Gross-Pitaevskii equation [Eq. (6) below] can be viewed as a coherent state in the total number of particles in the ground state such that the expectation value of the ground state number operator is N 0. The deviation from this number, however, is nonzero. In other words, N 0 = ˆN 0 2 ˆN It is precisely this indeterminacy in the number that leads to the concept of the phase of the condensate and is, in fact, the variable conjugate to the number. As such, their uncertainties obey a Heisenberg uncertainty principle. The approach I will present in Chap. 3, on the other hand, conserves the number of particles exactly leading to the interpretation of the solution of the Hartree-Fock equation, Eq. (18) or (25), as a number state (or Fock state). Further, in this number conserving approach, there are no physical consequences of the phase of the ground state of the bosonic system, and the expectation value of the ground state number operator is exactly N with no fluctuations. Since a coherent state can be constructed from number states, however, the solution to the Gross-Pitaevskii equation can be constructed from the Hartree-Fock

20 9 solutions for different numbers of particles. 2.1 Review of second quantization In the formalism of second quantization [44], the many-body wave function is transformed from configuration space to occupation number space. This transformation is accomplished via the independent particle representation and simplifies the construction of properly symmetrized many-body states within the independent particle approximation. This simplification is especially useful for keeping track of the combinatorial factors arising in the calculation of matrix elements for bosonic systems. Within this independent particle representation, a basis function in a many boson Hilbert space is specified by a set of occupation numbers n={n α } where α represents all of the quantum numbers needed to label a single particle state from some single particle basis {ψ α (x)}. For instance, n = n 0, 0,..., 0, n i, 0,... (1) is a many boson basis function with n 0 bosons in the 0-th single particle state, n i bosons in the i-th single particle state, and no bosons in any other single particle state. It is orthogonal to all other many-body basis states having different sets of occupation numbers, and it is normalized. In other words, n n =δ n,n. An additional consequence of this condition is that states with a different total number of particles are orthogonal. Taken together, all such many-body states form a complete expansion basis which spans the many-body Hilbert space for a given total number of particles. Of course, the sum of occupation numbers for any single many-body basis state necessarily equals the total number of particles, i.e. can also be written as n = (ĉ 0 )n0 n0! (ĉ i )ni ni! 0. i n i = N. The above basis state Here, 0 is the state with no bosons present in any single particle state. The creation, ĉ α, and annihilation, ĉ α, operators create and annihilate a boson in the α-th single particle state in the

21 10 following sense: ĉ α..., n α,... = n α , n α + 1,... ĉ α..., n α,... = n α..., n α 1,.... Further, the ĉ s satisfy the boson commutation relations [ ] ĉ α, ĉ β = δ αβ and [ ] ĉ α, ĉ β = [ĉ α, ĉ β ] = 0. Yet another way to write the basis state in Eq. (1), adopting now the coordinate representation, is n0!n i! Ψ n (x 1,..., x N ) = S [ψ 0 (x 1 ) ψ 0 (x n0 )ψ i (x n0+1) ψ i (x N )], N! with S the symmetrization operator. This form explicitly utilizes the configuration space viewpoint that is more commonly adopted in the context of atomic structure calculations. The Hamiltonian for a system of bosons interacting via two-body forces can be written in second quantization as Ĥ = αβ ĉ α α H 0 β ĉ β + 1 ĉ 2 αĉ β αβ V γδ ĉ δĉ γ. (2) αβγδ In this expression, all indices are summed over the complete set of single particle states and α H 0 β = d 3 x ψα(x) H 0 (x) ψ β (x) is the matrix element of H 0 (x) between single particle orbitals. All one-body operators such as the kinetic energy are included in H 0 (x) (see, for example, Eq. (16) in the next chapter). The two particle interaction matrix element in Eq. (2) is given by αβ V γδ = d 3 x d 3 x ψα(x) ψβ(x ) V (x x ) ψ γ (x) ψ δ (x ). The factor of 1/2 preceding the two-body interaction energy in Eq. (2) eliminates double counting of interacting pairs [50, 51]. The ψ α (x) in these matrix elements are (arbitrary) single particle orbitals that form a complete orthonormal basis: d 3 x ψα(x) ψ β (x) = δ αβ.

22 Bogoliubov approximation To better understand the differences and similarities between my formulation of Hartree-Fock and the random phase approximation (RPA) for bosons and the Bogoliubov approach more commonly used for boson systems, I reproduce here the basics of the conventional Bogoliubov derivation for a general two-body interaction. The essence of the Bogoliubov approximation lies in treating the condensate (i.e. the ground state configuration) separately from the rest of the system, an approximation justified by the comparatively small occupation of the excited configurations relative to the condensate configuration. In the limit N, the Bogoliubov approximation is exact as it is for noninteracting particles in the low temperature limit. For finite interacting systems, the assumption is that the condensate has on the order of N particles while the excited states collectively have on the order of 1 particle. Having made this approximation, an effective Hamiltonian is derived which has a quadratic dependence on excitation or fluctuation operators. This quadratic form can be diagonalized through the use of a canonical transformation [44]. Conventional approaches for bosons describe the state of the system in terms of fields rather than sets of occupation numbers [44]. The field operator ˆψ(x) ( ˆψ (x)) is defined as ˆψ(x) = α φ α (x)ĉ α where ĉ (ĉ ) is as before and φ α (x) is an arbitrary single particle basis function. The field operator ˆψ(x) ( ˆψ (x)) is interpreted as destroying (creating) a particle at a point x. Inverting the above relation and substituting it into Eq. (2) yields the following form for the Hamiltonian in terms of the field operators [44]: Ĥ = d 3 x ˆψ (x) H 0 (x) ˆψ(x) d 3 x d 3 x ˆψ (x) ˆψ (x ) V (x x ) ˆψ(x) ˆψ(x ). (3) The standard Bogoliubov approach [44] separates the condensate from the excited states in the field operator (and its adjoint), ˆψ(x) = φ 0 (x)ĉ 0 + α 0 φ α (x)ĉ α,

23 12 replaces the operators ĉ 0 (ĉ 0 ) by the c-number N 0, and collects the sum over excited states into a fluctuation operator ˆϕ(x) ( ˆϕ (x)), ˆϕ(x) = α 0 φ α (x)ĉ α. Physically, this operator annihilates (creates) a particle in a singly excited state at position x and must be small, in some sense, compared to the condensate wave function in order to justify the expansion of the Hamiltonian only through quadratic terms in ˆϕ. The total field operator is then just ˆψ(x) N 0 φ 0 (x) + ˆϕ(x). (4) The consequence of this replacement is that the number operator, ˆN = d 3 x ˆψ (x) ˆψ(x) = N 0 + N 0 d 3 x ( φ 0(x) ˆϕ(x) + ˆϕ (x)φ 0 (x) ) + d 3 x ˆϕ (x) ˆϕ(x), no longer commutes with the Hamiltonian so that the number of particles is not conserved. This shortcoming can be approximately overcome by instead using the grand canonical Hamiltonian, ˆK = Ĥ µ ˆN [44]. In this expression, µ is the chemical potential which is chosen to fix the average number of particles. Excitation energies can be computed directly within this approach, but it has recently been pointed out [52] that the fact that the number of particles is not conserved implies the existence of a solution of the normal mode equations with a vanishing excitation energy the Goldstone mode [14, 54] that restores the lost symmetry. Such symmetryrestoring modes are common in RPA analyses and arise whenever a continuous symmetry present in the Hamiltonian is broken for any reason in the mean field solution [45, 53, 66]. After the field operator from Eq. (4) is substituted into ˆK and terms through O( ˆϕ 2 ) are retained, ˆK can be diagonalized with the canonical transformation [84, 44] ˆϕ(x) = λ u λ (x) ˆβ λ + v λ(x) ˆβ λ. (5)

24 13 The ˆβ λ ( ˆβ λ ) are interpreted as annihilation (creation) operators for quasi-particles. Diagonalizing ˆK in this approximation is thus equivalent to transforming to a system of non-interacting quasi-particles [see Eq. (50)]. ˆK can only be diagonalized, however, provided the condensate wave function φ 0 (x) satisfies the self-consistent equation [H 0 (x)+n 0 ] d 3 x φ 0(x )V (x x )φ 0 (x ) φ 0 (x)=µφ 0 (x). (6) This condition eliminates terms linear in ˆϕ(x) from ˆK defining its minimum with respect to small fluctuations. Equation 6 is known as the nonlinear Schrödinger equation or the Gross- Pitaevskii equation [55]. Given the interpretation of ˆϕ(x), it is evident that the elimination of linear terms in ˆϕ from ˆK builds single particle excitations into the condensate wave function. Keeping terms through O( ˆϕ 2 ) in ˆK allows for only single and double excitations of the system state ket. In the next chapter, I will present a number conserving theory that includes much the same physics, but which starts from a very different point of view. For instance, the Hartree- Fock approximation (see Sec. 3.1) includes single particle excitations and results in an equation similar to Eq. (6). The RPA (see Sec. 3.3) encompasses essentially the same physics of double excitations as the expansion of ˆK to second order in ˆϕ. After the Bogoliubov transformation, the grand canonical Hamiltonian takes the simple form ˆK = E 0 µn 0 λ hω λ d 3 x vλ(x)v λ (x) + λ hω λ ˆβ λ ˆβ λ (7) provided u λ (x) and v λ (x) satisfy the normal mode equations ] [H 0 (x) µ + N 0 d 3 x φ 0(x ) V (x x ) φ 0 (x ) u λ (x) + N 0 d 3 x φ 0(x ) V (x x ) u λ (x ) φ 0 (x) + N 0 d 3 x φ 0 (x ) V (x x ) v λ (x ) φ 0 (x) = hω λ u λ (x) ] [H 0 (x) µ + N 0 d 3 x φ 0(x ) V (x x ) φ 0 (x ) v λ (x) + N 0 d 3 x φ 0 (x ) V (x x ) v λ (x ) φ 0(x) + N 0 d 3 x φ 0(x ) V (x x ) u λ (x ) φ 0(x) = hω λ v λ (x). (8)

25 14 In these equations, hω λ is the excitation energy, and the solutions u λ (x) and v λ (x) are normalized as d 3 x u λ (x)u λ(x) vλ (x)v λ(x) = δ λ λ in order to preserve the bosonic commutation relations of ˆβ and ˆϕ. The normal mode equations can be additionally transformed into algebraic eigenvalue equations by expanding u(x) and v(x) on a single particle basis. Specifically, u λ (x) = p 0 U λp φ p (x) v λ (x) = p 0 V λp φ p (x). A convenient and physical choice for the single particle basis is the set of states that satisfy H 0 (x)φ i (x) + N [ 0 d 3 x φ 2 0(x ) V (x x ) φ 0 (x ) φ i (x) + ] d 3 x φ 0(x ) V (x x ) φ i (x ) φ 0 (x) = ɛ i φ i (x). where ɛ 0 =µ. This basis is physically sensible since it includes the mean field effects of the condensate. Using such a basis to solve the normal mode equations usually requires fewer often far fewer states than would be needed if u λ (x) and v λ (x) were expanded in terms of a harmonic oscillator basis as is one common method of solution [11]. N 0 2 N 0 2 [ Uλp q0 V p0 + V λp qp V 00 ] + (ɛ q µ) U λq = hω λ U λq p 0 [ ] Uλp qp V 00 + Vλp q0 V p0 + (ɛq µ) V λq = hω λ V λq. (9) p 0 The shift in the ground state energy [see Eq. (7)] can now be written in the {φ i (x)} representation as hω λ d 3 x vλ(x)v λ (x) = λ λ hω λ V λp 2. (10) p 0

26 Thomas-Fermi approximation The Thomas-Fermi approximation greatly simplifies the solution of the Gross- Pitaevskii equation, Eq. (6), by reducing it to an algebraic one. This reduction is accomplished by neglecting the kinetic energy and retaining the trapping potential and the mean field interaction. It follows that this approximation is valid in the limit that the mean field term dominates. To make the approximation more concrete, consider the Gross-Pitaevskii equation in the shape independent approximation (see Sec. 3.5) scaled by the harmonic oscillator energy, hω, and length, β= h/mω: [ ( ω 2 x x 2 + ω yỹ ω z 2 z 2) ] + α φ 0 ( x) 2 φ 0 ( x) = µφ 0 ( x). (11) In this expression, α=4πn 0 ã sc with a sc the scattering length, and the tildes denote rescaled quantities, i.e. µ= hω µ and x=β x. The reference frequency ω entering the scaling is chosen to be some convenient value. Note that the two-body scattering information and the number dependence enter only through the parameter α. The validity condition for the Thomas-Fermi approximation can now be written as α 1. This condition must be supplemented by the caveat that the local kinetic energy (which has been neglected) should also be small compared to the mean field energy. For simplicity, I consider an isotropic oscillator. The method can, of course, be applied to anisotropic systems and even multiple component systems [13, 24], but the additional algebraic complication is not particularly enlightening. So, neglecting kinetic energy, Eq. 11 can be rearranged to give the Thomas-Fermi approximation to the probability density, 1 ( µ φ 0 (x) 2 2 = r2) /α, 0 r 2 µ 0, r > 2 µ (12) The limit in r arises from the requirement that the probability density be a non-negative number. It is in the region of this upper limit in r that the Thomas-Fermi approximation will break down since the local kinetic energy dominates the mean field. In practice, however, this failure is not

27 16 a significant drawback. The remaining free parameter, µ, is chosen so that the wave function is normalized. The result is ( 15α µ = 4 2 ) 2 5 (13) The Thomas-Fermi approximation for bosons in a trap has a simple interpretation in the sense that the bosons can be imagined to be a liquid filling the bottom of the trap up to the level µ. The resulting effective potential the combination of the trapping potential and the mean field is thus a flat bottomed version of the potential. The analogy to liquid drops has been pursued with considerable success to calculate, for instance, the low-lying excitation spectrum of a condensate [19, 22]. An especially attractive feature of the approximation is that nearly all quantities can be calculated analytically, which yields useful scaling laws in the limit α 1.

28 CHAPTER 3 ZERO TEMPERATURE THEORY FOR SINGLE COMPONENT CONDENSATES The recent experimental observations of Bose-Einstein condensates [1, 2, 3, 4, 5, 6, 7] and the successful experiments [9, 10] on condensate properties have increased the desirability of a comprehensive theoretical formulation. Several groups [11, 19, 32, 52, 56, 57, 58, 59] have made progress in this direction by adopting the standard Bogoliubov approach for many interacting bosons [44]. This is an approach that treats the condensate as a reservoir which can exchange both particles and energy with the rest of the system. This approximation, however, does not inherently conserve the number of particles, although the chemical potential µ can be introduced to enforce this condition on average. In order to connect to many-body approaches such as those used in atomic structure calculations, I formulate the theory for trapped atoms using standard Schrödinger quantum mechanics [60]. This, of course, automatically conserves the number of interacting particles. This methodology pursues the analogy of atoms in a trap to electrons trapped by the Coulomb field of a nucleus. A fundamental difference between these cases is, of course, the character of the particles: the atoms experimentally studied in such traps to date are bosons, whose exchange properties differ simply yet profoundly from the fermionic electrons in an atom. This viewpoint allows concepts such as quasi-particles to be discussed in terms of configurations and orbitals, and permits the language of condensed matter physics to be linked to that of atomic physics and nuclear physics. As I will show below, this formulation leads to results which are largely equivalent to those obtained in the Bogoliubov approach, aside from very minor differences that should be unimportant for current experimental conditions. A key byproduct of this approach is that it permits the application of standard tools of atomic theory, such as configuration

29 18 interaction [61], which transcend Hartree-Fock theory in order to describe new phenomena such as multiple particle excitations that are not encompassed by Bogoliubov theory. 3.1 Hartree-Fock approximation With the Hartree-Fock approximation, one seeks the best independent particle wave function given the occupancy of each single particle orbital. In the present case, I concentrate on the ground state of a system of bosons (i.e. all particles occupy the lowest orbital) although more general occupation schemes can be used. In such cases, however, basic properties of the single particle states such as orthogonality must be explicitly addressed. Considerable freedom exists in the choice of a single particle basis set. This flexibility is used to derive an equation that determines those single particle states which variationally minimize the total energy. In other words, the Hamiltonian is approximately diagonalized, including as much of the interparticle interactions as is possible given that the trial wave function is constrained to independent particle form. The Hartree-Fock equation can be derived from either first- or second-quantized formalisms. Each provides separate and useful insights. The first-quantized derivation provides a simple picture that can be easily understood in terms of basic quantum mechanics. The secondquantized derivation, on the other hand, provides greater insight into the physics included in the trial wave function. Both approaches, of course, yield identical results and will be described below First-quantized approach. The derivation here is first-quantized in that the many-body wave function is expressed directly in terms of spatial coordinates, although it is written in the independent particle approximation. In many respects, the first-quantized derivation presented here parallels the first-quantized derivation of the Hartree-Fock equations for fermions (see Cowan [61], for example). The ansatz for the total ground state wave function Φ in the independent particle approximation is expressed as Φ(x 1,..., x N ) ψ 0 (x 1 )... ψ 0 (x N ) (14)

30 19 where the single particle orbitals ψ 0 are to be determined. The spin part of the wave function is similarly a product of the spin kets for each atom and otherwise does not enter the calculation. The equation for ψ 0 is obtained by applying the variational principle to the Hamiltonian N N H = H 0 (x i ) + V (x i x j ). (15) i=1 In this expression, the one particle operator H 0 (x) includes any external trapping potential V ext (x) and is given by i<j H 0 (x) = h2 2m 2 + V ext (x) (16) For typical magnetic traps V ext (x) is a cylindrically symmetric harmonic trapping potential; in atomic structure calculations, V ext (x) is the electron-nucleus Coulomb interaction. The two particle operator V (x i x j ) in Eq. (15) is the particle-particle interaction. In the case of neutral trapped atoms, it is a potential with a strongly repulsive core at distances of roughly a few atomic units, a well at a few tens of atomic units, and a van der Waals tail. In atomic structure calculations, the particle-particle interaction is just the electronic Coulomb repulsion. Given the trial wave function in Eq. (14), the expectation value of the Hamiltonian [Eq. (15)] the total energy for this system of N particles is then E0 HF = N ψ 0 H 0 ψ 0 + ψ 0 ψ 0 This notation for the one particle matrix element is interpreted as N(N 1) ψ 0 ψ 0 V ψ 0 ψ 0 2 ψ 0 ψ 0 2. (17) ψ 0 H 0 ψ 0 = d 3 x ψ0(x) H 0 (x) ψ 0 (x) while the notation for the two particle matrix element implies a double integral over all coordinates of two particles: ψ 0 ψ 0 V ψ 0 ψ 0 = d 3 x d 3 x ψ0(x) ψ0(x ) V (x x ) ψ 0 (x) ψ 0 (x ). Taking the variation of E with respect to ψ 0 gives, after some algebra, δe0 HF = N δψ 0 H 0 ψ 0 + ψ 0 ψ 0 N(N 1) 2 δψ 0 ψ 0 + ψ 0 δψ 0 V ψ 0 ψ 0 ψ 0 ψ 0 2

31 20 N ( E HF 0 N (N 1) 2 ) ψ 0 ψ 0 V ψ 0 ψ 0 δψ 0 ψ 0 ψ 0 ψ 0 2 ψ 0 ψ 0. If arbitrary variations δψ 0 are allowed, the first variation δe HF 0 =0 subject to ψ 0 ψ 0 =1 is satisfied when [ ] H 0 (x)+(n 1) d 3 x ψ0(x ) V (x x ) ψ 0 (x ) ψ 0 (x)= ε 0 ψ 0 (x). (18) For a system of only one boson, Eq. (18) reduces to the appropriate noninteracting Schrödinger equation. Moreover, this equation is the number conserving analogue of the nonlinear Schrödinger equation for the condensate wave function (see Ref. [44, 55, 62] and Eq. (6) in Sec. 2.2). The eigenenergy ε 0 in Eq. (18) is defined as ε 0 = EHF 0 N + (N 1) 2 ψ 0 ψ 0 V ψ 0 ψ 0. (19) Being the eigenvalue of the equation for the ground state orbital, ε 0 is the ground state orbital energy. Interestingly, ε 0 obeys Koopmans theorem [63] as do the orbital energies for fermions. The statement of Koopmans theorem applicable to a system of bosons is that the orbital energy represents the difference between the Hartree-Fock ground state energy for N particles and N 1 particles provided the difference between the ground state orbital for N particles and the ground state orbital for N 1 orbitals can be neglected. In the limit N 1, the latter approximation is physically reasonable given the order N 1 effect of a single additional particle on the orbital. In fact, this approximation holds quite well for as few as 10 particles. From the above statement, it can also be recognized that Koopmans theorem is essentially a statement of the definition of the chemical potential encountered in the Bogoliubov approach (see Sec. 2.2). From Eq. (19), the total energy for a system of N particles can be written as E0 HF N(N 1) = N ψ 0 H 0 ψ 0 + ψ 0 ψ 0 V ψ 0 ψ 0. 2 The energy difference between a system with N particles and one with N 1 is thus E HF 0 (N) E HF 0 (N 1) = ψ 0 H 0 ψ 0 + (N 1) ψ 0 ψ 0 V ψ 0 ψ 0 = ε 0. Thus, Koopmans theorem is also satisfied by bosons.